We can pass to the local ring \(S\)  that exists by Remark REF  and assume \(R\)  is F-finite, i.e., \({{}^{\varphi ^n}\!}R \in \operatorname{mod}(R)\) .
Set \(\ell =d-n\) . Then \(\ell \ge 1\) . Also, by (ii) and the Theorem REF , we have
\(\operatorname{Ext}^{\ell }_R(M\otimes _R{{}^{\varphi ^r}\!}R,R)=0.\qquad \mathrm {(\ref {tt}.1)}\) 
Let \(\mathfrak {p}\in \operatorname{Supp}_R(\operatorname{Tor}_i^R(M,{{}^{\varphi ^r}\!}R))\)  for some \(i\ge 1\) . It follows from (i) that
\(\operatorname{depth}(R_{\mathfrak {p}})=\dim (R_{\mathfrak {p}})\ge d-n>d-n-i.\qquad \mathrm {(\ref {tt}.2)}\) 
Hence, by (REF .1) and (REF .2), we deduce that
\(\operatorname{\textnormal {grade}}_R(\operatorname{Tor}_i^R(M,{{}^{\varphi ^r}\!}R))>d-n-i=\ell -i \text{ for all } i \text{ with } 1\le i\le \ell .\qquad \mathrm {(\ref {tt}.3)}\) 
As \(R\)  is Gorenstein, it follows \({{}^{\varphi ^r}\!}R\cong \operatorname{Hom}_R({{}^{\varphi ^r}\!}R,R)\)  [1]}. Also, since \({{}^{\varphi ^r}\!}R\)  is a maximal Cohen-Macaulay \(R\) -module, we see \(\operatorname{Ext}^j_R({{}^{\varphi ^r}\!}R,R)=0\)  for all \(j\ge 1\) . Hence, in view of Lemma REF (ii), (REF .1) and (REF .3) we conclude that
\(\operatorname{Ext}^{\ell }_R(M,{{}^{\varphi ^r}\!}R)=\operatorname{Ext}^{\ell }_R(M,\operatorname{Hom}_R({{}^{\varphi ^r}\!}R,R))=0.\) 
Now, by the four-term exact sequence (REF .4) we obtain \(\operatorname{Tor}_1^R(\operatorname{Tr}\Omega ^{\ell }M,{{}^{\varphi ^r}\!}R)=0\) . So [2]} implies that \(\operatorname{Tor}_i^R(\operatorname{Tr}\Omega ^{\ell }M,{{}^{\varphi ^r}\!}R)=0\)  for all \(i\ge 1\)  and \(\operatorname{pd}_R(\operatorname{Tr}\Omega ^{\ell }M)<\infty \) . Another use of (REF .4) implies that \(\operatorname{Ext}^{\ell }_R(M,R)=0\) . On the other hand, as \(M\)  satisfies \((S_n)\) , it is an \(n\) -th syzygy module; see [3]}. Therefore, \(\Omega ^{\ell }M\)  is a \(d\) -th syzygy module, or equivalently, is a maximal Cohen-Macaulay module. This implies that \(\operatorname{Tr}\Omega ^{\ell }M\)  is also maximal Cohen-Macaulay. As \(\operatorname{Tr}\operatorname{Tr}(-)\approx (-)\) , we deduce that \(\Omega ^{\ell }M\)  is free and \(\operatorname{pd}_R(M)\le \ell \) . Consequently, the fact that \(\operatorname{Ext}^{\ell }_R(M,R)=0\)  yields \(\operatorname{pd}_R(M)<\ell =d-n\) , as required.
